Overview
Percentages and Percent Change is among the most frequently tested topics in the Problem-Solving and Data Analysis domain on the digital SAT. Questions require converting between fractions, decimals, and percents; computing the percent of a number; applying the percent change formula; and handling real-world applications such as markup, discount, sales tax, tip, and commission. These are calculator-active questions, and the topic spans both easy and medium difficulty levels.
Key Points
1. Converting Between Fractions, Decimals, and Percents
| From | To | Operation |
|---|---|---|
| Percent | Decimal | Divide by 100 (move decimal left 2 places) |
| Decimal | Percent | Multiply by 100 (move decimal right 2 places) |
| Fraction | Decimal | Divide numerator by denominator |
| Decimal | Fraction | Write over appropriate power of 10 and simplify |
Examples: 75% = 0.75 = 3/4; 0.08 = 8% = 2/25
2. Percent of a Number
Percent of a number: part = (percent / 100) × whole
Or using decimal: part = decimal × whole
Example: 35% of 240 = 0.35 × 240 = 84
3. Percent Change Formula
- Positive result = percent increase
- Negative result = percent decrease
- The original (before-change) value is always the denominator
Multiplier form: For an r% increase → multiply by (1 + r/100); for r% decrease → multiply by (1 − r/100)
| Change | Multiplier |
|---|---|
| 20% increase | × 1.20 |
| 15% decrease | × 0.85 |
| 100% increase (doubles) | × 2.00 |
4. Reverse Percent (Finding the Original)
“After a 30% increase, the price is $130. What was the original price?”
- original × 1.30 = 130 → original = 130 / 1.30 = $100
Never subtract 30% from 91, not $100).
5. Successive Percent Changes
Never add successive percent changes. Always multiply the multipliers.
Examples:
- 20% increase then 20% decrease: 1.20 × 0.80 = 0.96 → 4% net decrease (NOT 0%)
- 10% increase then 10% increase: 1.10 × 1.10 = 1.21 → 21% total increase (not 20%)
“Plug 100” method: Start with 100, apply each multiplier in sequence, compare the result to 100 to find the net change.
6. Real-World Applications
| Application | Formula |
|---|---|
| Markup | Selling price = Cost × (1 + markup rate) |
| Discount | Sale price = Original × (1 − discount rate) |
| Sales tax | Final price = Pre-tax × (1 + tax rate) |
| Tip | Total = Bill × (1 + tip rate) |
| Commission | Commission = Sales amount × commission rate |
| Percent error | |Measured − Actual| / Actual × 100% |
Pitfalls and Common Mistakes
Mistake 1: Adding successive percent changes instead of multiplying multipliers. A 15% increase followed by a 10% decrease is NOT a net 5% increase. Fix: Convert each change to a multiplier and multiply: 1.15 × 0.90 = 1.035 → 3.5% net increase.
Mistake 2: Using the new value as the base for percent change. “A price dropped from 60. What is the percent decrease?” Students compute (80−60)/60 = 33.3% instead of (80−60)/80 = 25%. Fix: Percent change always divides by the original (starting) value, never the new value.
Mistake 3: Subtracting a percent from the new value to find the original. “After a 25% increase the price is 56.25 instead of 75 / 1.25 = $60. Fix: Set up original × multiplier = new, then divide both sides by the multiplier.
Mistake 4: Confusing percent increase with doubling. A 100% increase doubles the value; a 200% increase triples it. “200% of” and “200% more than” are different. Fix: “200% of X” = 2X; “200% more than X” = X + 2X = 3X.
Related Entries
Quick Reference Card
| Concept | Formula |
|---|---|
| Percent of a number | part = (percent/100) × whole |
| Percent change | (new − original) / original × 100% |
| Percent increase multiplier | 1 + r (e.g., 20% → 1.20) |
| Percent decrease multiplier | 1 − r (e.g., 15% → 0.85) |
| Successive changes | Multiply multipliers: m₁ × m₂ × … |
| Reverse percent | original = new ÷ multiplier |
| Percent error | |measured − actual| / actual × 100% |